## Journal of Commutative Algebra

### Lattice complements and the subadditivity of syzygies of simplicial forests

Sara Faridi

#### Abstract

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals of simplicial forests. We do this by interpreting the subadditivity property for any monomial ideal as a property of homologies of its lcm lattice. For an ideal $I$ that is the facet ideal of a simplicial forest, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti number. The subadditivity formula follows from this fact.

#### Article information

Source
J. Commut. Algebra, Volume 11, Number 4 (2019), 535-546.

Dates
First available in Project Euclid: 7 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.jca/1575687733

Digital Object Identifier
doi:10.1216/JCA-2019-11-4-535

Mathematical Reviews number (MathSciNet)
MR4039981

Zentralblatt MATH identifier
07147395

Subjects
Primary: 13D02: Syzygies, resolutions, complexes

#### Citation

Faridi, Sara. Lattice complements and the subadditivity of syzygies of simplicial forests. J. Commut. Algebra 11 (2019), no. 4, 535--546. doi:10.1216/JCA-2019-11-4-535. https://projecteuclid.org/euclid.jca/1575687733

#### References

• L. Avramov, A. Conca and S. Iyengar, Subadditivity of syzygies of Koszul algebras, Math. Ann. \bff 361 (2015), no. 1-2, 511–534. @articleACI, MRKEY = MR3302628, AUTHOR = Avramov, Luchezar L. and Conca, Aldo and Iyengar, Srikanth B., TITLE = Subadditivity of syzygies of Koszul algebras, JOURNAL = Math. Ann., FJOURNAL = Mathematische Annalen, VOLUME = 361, YEAR = 2015, NUMBER = 1-2, PAGES = 511–534, ISSN = 0025-5831, MRCLASS = 13D07 (13D02), MRNUMBER = 3302628, MRREVIEWER = Timothy B. P. Clark, DOI = 10.1007/s00208-014-1060-4, URL =,
• A. Abedelfatah and E. Nevo, On vanishing patterns in j-strands of edge ideals, arXiv:1510.08192 (2016). @articleAN, MRKEY = MR3680616, AUTHOR = Abedelfatah, Abed and Nevo, Eran, TITLE = On vanishing patterns in $j$-strands of edge ideals, JOURNAL = J. Algebraic Combin., FJOURNAL = Journal of Algebraic Combinatorics. An International Journal, VOLUME = 46, YEAR = 2017, NUMBER = 2, PAGES = 287–295, ISSN = 0925-9899, MRCLASS = 13A15 (05E40), MRNUMBER = 3680616, MRREVIEWER = Ashkan Nikseresht, DOI = 10.1007/s10801-017-0747-5, URL =, @techreportAN, TITLE = On vanishing patterns in $j$-strands of edge ideals, AUTHOR = Abedelfatah, Abed and Nevo, Eran, ARXIV = 1510.08192, YEAR = 2015,
• K. Baclawski, Galois connections and the Leray spectral sequence, Advances in Math. \bff 25 (1977), no. 3, 191–215. @articleB, MRKEY = MR470035, AUTHOR = Baclawski, Kenneth, TITLE = Galois connections and the Leray spectral sequence, JOURNAL = Advances in Math., FJOURNAL = Advances in Mathematics, VOLUME = 25, YEAR = 1977, NUMBER = 3, PAGES = 191–215, ISSN = 0001-8708, MRCLASS = 18G99 (06A10), MRNUMBER = 470035, MRREVIEWER = T. Y. Lin, DOI = 10.1016/0001-8708(77)90073-1, URL =,
• N. Erey and S. Faridi, Multigraded Betti numbers of simplicial forests, J. Pure Appl. Algebra \bff 218 (2014), 1800–1805. @articleEF1, MRKEY = MR3195409, AUTHOR = Erey, Nursel and Faridi, Sara, TITLE = Multigraded Betti numbers of simplicial forests, JOURNAL = J. Pure Appl. Algebra, FJOURNAL = Journal of Pure and Applied Algebra, VOLUME = 218, YEAR = 2014, NUMBER = 10, PAGES = 1800–1805, ISSN = 0022-4049, MRCLASS = 13F55 (05C05 05E40 05E45 13D02), MRNUMBER = 3195409, MRREVIEWER = Thomas Kahle, DOI = 10.1016/j.jpaa.2014.02.006, URL =,
• N. Erey and S. Faridi, Betti numbers of monomial ideals via facet covers, J. Pure Appl. Algebra \bff 220 (2016), no. 5, 1990–2000. @articleEF2, MRKEY = MR3437276, AUTHOR = Erey, Nursel and Faridi, Sara, TITLE = Betti numbers of monomial ideals via facet covers, JOURNAL = J. Pure Appl. Algebra, FJOURNAL = Journal of Pure and Applied Algebra, VOLUME = 220, YEAR = 2016, NUMBER = 5, PAGES = 1990–2000, ISSN = 0022-4049, MRCLASS = 13B02 (13F20), MRNUMBER = 3437276, MRREVIEWER = Driss Bennis, DOI = 10.1016/j.jpaa.2015.10.011, URL =,
• D. Eisenbud, C. Huneke and Bernd Ulrich, The regularity of Tor and graded Betti numbers, Amer. J. Math. \bff 128 (2006), no. 3, 573-605. @articleEHU, MRKEY = MR2230917, AUTHOR = Eisenbud, David and Huneke, Craig and Ulrich, Bernd, TITLE = The regularity of Tor and graded Betti numbers, JOURNAL = Amer. J. Math., FJOURNAL = American Journal of Mathematics, VOLUME = 128, YEAR = 2006, NUMBER = 3, PAGES = 573–605, ISSN = 0002-9327, MRCLASS = 13D45 (13D02), MRNUMBER = 2230917, MRREVIEWER = Graham J. Leuschke, URL = http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.3eisenbud.pdf,
• S. El Khoury and H. Srinivasan, A note on the subadditivity of syzygies, arXiv:1602.02116 (2016). @articleES, MRKEY = MR3661644, AUTHOR = El Khoury, Sabine and Srinivasan, Hema, TITLE = A note on the subadditivity of syzygies, JOURNAL = J. Algebra Appl., FJOURNAL = Journal of Algebra and its Applications, VOLUME = 16, YEAR = 2017, NUMBER = 9, PAGES = 1750177, 5, ISSN = 0219-4988, MRCLASS = 13D02 (13D07), MRNUMBER = 3661644, MRREVIEWER = Ştefan Ovidiu Tohǎneanu, DOI = 10.1142/S0219498817501778, URL =, @techreportES, TITLE = A note on the subadditivity of Syzygies, AUTHOR = Khoury, Sabine El and Srinivasan, Hema, ARXIV = 1602.02116, YEAR = 2016,
• S. Faridi, The facet ideal of a simplicial complex, Manuscripta Math. \bff 109 (2002), 159–174. @articleF1, MRKEY = MR1935027, AUTHOR = Faridi, Sara, TITLE = The facet ideal of a simplicial complex, JOURNAL = Manuscripta Math., FJOURNAL = Manuscripta Mathematica, VOLUME = 109, YEAR = 2002, NUMBER = 2, PAGES = 159–174, ISSN = 0025-2611, MRCLASS = 13F55 (05C05 05C65 13C15), MRNUMBER = 1935027, MRREVIEWER = Rafael H. Villarreal, DOI = 10.1007/s00229-002-0293-9, URL =,
• S. Faridi, A good leaf order on simplicial trees, pp. 191–207 in Connections between algebra, combinatorics, and geometry, Springer Proc. Math. Stat. 76, Springer, New York, 2014. @incollectionF2, MRKEY = MR3213520, AUTHOR = Faridi, Sara, TITLE = A good leaf order on simplicial trees, BOOKTITLE = Connections between algebra, combinatorics, and geometry, SERIES = Springer Proc. Math. Stat., VOLUME = 76, PAGES = 191–207, PUBLISHER = Springer, New York, YEAR = 2014, MRCLASS = 13D02 (05E45 13F55), MRNUMBER = 3213520, MRREVIEWER = Somayeh Moradi, DOI = 10.1007/978-1-4939-0626-0_4, URL =,
• O. Fernández-Ramos and P. Gimenez, Regularity 3 in edge ideals associated to bipartite graphs, J. Algebraic Combin. \bff 39 (2014), 919–937. @articleFG, MRKEY = MR3199032, AUTHOR = Fernández-Ramos, Oscar and Gimenez, Philippe, TITLE = Regularity 3 in edge ideals associated to bipartite graphs, JOURNAL = J. Algebraic Combin., FJOURNAL = Journal of Algebraic Combinatorics. An International Journal, VOLUME = 39, YEAR = 2014, NUMBER = 4, PAGES = 919–937, ISSN = 0925-9899, MRCLASS = 05E18, MRNUMBER = 3199032, DOI = 10.1007/s10801-013-0473-6, URL =,
• V. Gasharov, I. Peeva and V. Welker, The lcm-lattice in monomial resolutions, Math. Res. Lett. \bff 6 (1999), 521–532. @articleGPW, MRKEY = MR1739211, AUTHOR = Gasharov, Vesselin and Peeva, Irena and Welker, Volkmar, TITLE = The lcm-lattice in monomial resolutions, JOURNAL = Math. Res. Lett., FJOURNAL = Mathematical Research Letters, VOLUME = 6, YEAR = 1999, NUMBER = 5-6, PAGES = 521–532, ISSN = 1073-2780, MRCLASS = 13D02 (13D25), MRNUMBER = 1739211, MRREVIEWER = Philippe Gimenez, DOI = 10.4310/MRL.1999.v6.n5.a5, URL =,
• J. Herzog and H. Srinivasan, On the subadditivity problem for maximal shifts in free resolutions, pp. 245–249 in Commutative algebra and noncommutative algebraic geometry, II, MSRI Publications 68, Cambridge Univ. Press (2015). @incollectionHS, MRKEY = MR3496869, AUTHOR = Herzog, Jürgen and Srinivasan, Hema, TITLE = On the subadditivity problem for maximal shifts in free resolutions, BOOKTITLE = Commutative algebra and noncommutative algebraic geometry. Vol. II, SERIES = Math. Sci. Res. Inst. Publ., VOLUME = 68, PAGES = 245–249, PUBLISHER = Cambridge Univ. Press, New York, YEAR = 2015, MRCLASS = 13D02, MRNUMBER = 3496869, MRREVIEWER = Ali Akbar Yazdan Pour,
• T. Ha and A. Van Tuyl, Splittable ideals and the resolutions of monomial ideals, J. Algebra \bff 309 (2007), 405-425. @articleHV, MRKEY = MR2301246, AUTHOR = Hà, Huy Tài and Van Tuyl, Adam, TITLE = Splittable ideals and the resolutions of monomial ideals, JOURNAL = J. Algebra, FJOURNAL = Journal of Algebra, VOLUME = 309, YEAR = 2007, NUMBER = 1, PAGES = 405–425, ISSN = 0021-8693, MRCLASS = 13D02 (13F55), MRNUMBER = 2301246, MRREVIEWER = Xinxian Zheng, DOI = 10.1016/j.jalgebra.2006.08.022, URL =,
• I. Peeva, Graded syzygies, Algebra and Applications 14, Springer (2011). @bookP, MRKEY = MR2560561, AUTHOR = Peeva, Irena, TITLE = Graded syzygies, SERIES = Algebra and Applications, VOLUME = 14, PUBLISHER = Springer-Verlag London, Ltd., London, YEAR = 2011, PAGES = xii+302, ISBN = 978-0-85729-176-9, MRCLASS = 13D02 (05E40 13D40 13F55 14M25), MRNUMBER = 2560561, MRREVIEWER = Christopher A. Francisco, DOI = 10.1007/978-0-85729-177-6, URL =,